Unbiased Instrumental Variables Estimation Under Known First-Stage Sign

Armstrong, Timothy B.; Andrews, Isaiah

doi:10.2139/ssrn.2673587

Cited by 10 publications

(14 citation statements)

References 31 publications

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“…10 Distinct but related are the impossibility results by Dufour (1997) and Hirano and Porter (2015); their setup is a generalization of the weak linear IV structure, whereas our setup is a generalization of the weak identification phenomenon. Indeed, Dufour (1997) showed nonexistence of bounded confidence sets (which is “stronger” than nonexistence of consistent estimators) while there exist weakly regular parameters that admit bounded confidence sets; 11 Hirano and Porter (2015) showed the impossibility of unbiased estimation while there exist weakly regular parameters that admit unbiased estimation (Andrews and Armstrong (2017)).…”

Section: Weak Identification In Semiparametric Modelsmentioning

confidence: 99%

“…Example We verify regularity of 2SLS, optimal IV, GMM, limited information maximum likelihood (LIML), continuously updating GMM (CUE), Fuller, and unbiased (Andrews and Armstrong (2017)) estimators. Observe that the reduced‐form coefficients

false(γ, π false)

are regular and the 2SLS can be written as a function of their estimators

{\overset{π}{true}}_{n} = {false(Z^{'} Z false)}^{−1} Z^{'} X

and

{\overset{γ}{true}}_{n} = {false(Z^{'} Z false)}^{−1} Z^{'} Y

{\overset{β}{true}}_{2SLS} = {true2.4ex2.4ex({\overset{π}{true}}_{n}^{'} true2.4ex2.4ex(Z^{'} Z true2.4ex2.4ex) {\overset{π}{true}}_{n} true2.4ex2.4ex)}^{−1} {\overset{π}{true}}_{n}^{'} true2.4ex2.4ex(Z^{'} Z true2.4ex2.4ex) {\overset{γ}{true}}_{n} = {true2.4ex2.4ex(\sqrt{n} {\overset{π}{true}}_{n}^{'} double-struckE true2.4ex2.4ex[z z^{'} true2.4ex2.4ex] \sqrt{n} {\overset{π}{true}}_{n} true2.4ex2.4ex)}^{−1} \sqrt{n} {\overset{π}{true}}_{n}^{'} double-struckE true2.4ex2.4ex[z z^{'} true2.4ex2.4ex] \sqrt{n} {\overset{γ}{true}}_{n} + o_{P} false(1 false) .

The residual is

o_{P} false(1 false)

since

false(Z^{'} Z false) false/ n

converges to

double-struckE false[z z^{'} false]

in probability under every path.…”

Section: Weak Efficiency For Weakly Regular Parametersmentioning

confidence: 99%

See 1 more Smart Citation

Theory of Weak Identification in Semiparametric Models

Kaji

2021

ECTA

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We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, that is, when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficient, it introduces noise in the corresponding estimation of weakly regular parameters, whence we can improve the estimators by local asymptotic Rao–Blackwellization. We call an estimator weakly efficient if it does not admit such improvement. New weakly efficient estimators are presented in linear IV and nonlinear regression models. Simulation of a linear IV model demonstrates how 2SLS and optimal IV estimators are improved.

show abstract

Section: Weak Identification In Semiparametric Modelsmentioning

confidence: 99%

false(γ, π false)

are regular and the 2SLS can be written as a function of their estimators

{\overset{π}{true}}_{n} = {false(Z^{'} Z false)}^{−1} Z^{'} X

and

{\overset{γ}{true}}_{n} = {false(Z^{'} Z false)}^{−1} Z^{'} Y

{\overset{β}{true}}_{2SLS} = {true2.4ex2.4ex({\overset{π}{true}}_{n}^{'} true2.4ex2.4ex(Z^{'} Z true2.4ex2.4ex) {\overset{π}{true}}_{n} true2.4ex2.4ex)}^{−1} {\overset{π}{true}}_{n}^{'} true2.4ex2.4ex(Z^{'} Z true2.4ex2.4ex) {\overset{γ}{true}}_{n} = {true2.4ex2.4ex(\sqrt{n} {\overset{π}{true}}_{n}^{'} double-struckE true2.4ex2.4ex[z z^{'} true2.4ex2.4ex] \sqrt{n} {\overset{π}{true}}_{n} true2.4ex2.4ex)}^{−1} \sqrt{n} {\overset{π}{true}}_{n}^{'} double-struckE true2.4ex2.4ex[z z^{'} true2.4ex2.4ex] \sqrt{n} {\overset{γ}{true}}_{n} + o_{P} false(1 false) .

The residual is

o_{P} false(1 false)

since

false(Z^{'} Z false) false/ n

converges to

double-struckE false[z z^{'} false]

in probability under every path.…”

Section: Weak Efficiency For Weakly Regular Parametersmentioning

confidence: 99%

Theory of Weak Identification in Semiparametric Models

Kaji

2021

ECTA

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show abstract

“…Also, while we focus on the popular F-test as our pre-test, one advantage of our method versus Moreira (2009) is that it can be modified to handle a wide variety of pre-tests, such as (i) a pre-test based on checking whether the estimated signs of the regression coefficients between the instruments and the treatment are congruent to what's expected from subject-matter knowledge (Andrews and Armstrong, 2017), (ii) a pre-test based on applying a Lasso-type optimization to select the strongest instruments (Belloni et al, 2012), and (iii) a pre-test where the instruments are selected based on forward stepwise regression; see Section 4.2 for details.…”

Section: Prior Work and Our Contributionsmentioning

confidence: 99%

Inferring Treatment Effects After Testing Instrument Strength in Linear Models

Bi¹,

Kang²,

Taylor³

2020

Preprint

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A common practice in IV studies is to check for instrument strength, i.e. its association to the treatment, with an F-test from regression. If the F-statistic is above some threshold, usually 10, the instrument is deemed to satisfy one of the three core IV assumptions and used to test for the treatment effect. However, in many cases, the inference on the treatment effect does not take into account the strength test done a priori. In this paper, we show that not accounting for this pretest can severely distort the distribution of the test statistic and propose a method to correct this distortion, producing valid inference. A key insight in our method is to frame the F-test as a randomized convex optimization problem and to leverage recent methods in selective inference. We prove that our method provides conditional and marginal Type I error control. We also extend our method to weak instrument settings. We conclude with a reanalysis of studies concerning the effect of education on earning where we show that not accounting for pre-testing can dramatically alter the original conclusion about education's effects. | INTRODUCTION | MotivationInstrumental variables (IV) is a commonly used approach in economics, epidemiology, genetics, and health policy to estimate the effect of an exposure, treatment, or policy on an outcome of interest; see Angrist and Krueger (2001), Robins (2006), andBaiocchi et al. (2014) for overviews. IV methods require finding variables, known as 1

show abstract

“…13 Distinct but related are the impossibility results by Dufour (1997) and Hirano and Porter (2015); their setup is a generalization of the weak linear IV structure whereas our setup is a generalization of the weak identification phenomenon. Indeed, Dufour (1997) shows nonexistence of bounded confidence sets (which is "stronger" than nonexistence of consistent estimators) while there exist weakly regular parameters that admit bounded confidence sets; 14 Hirano and Porter (2015) show the impossibility of unbiased estimation while there exist weakly regular parameters that admit unbiased estimation (Andrews and Armstrong, 2017).…”

Section: Fundamental Impossibilitymentioning

confidence: 99%

Theory of Weak Identification in Semiparametric Models

Kaji

2019

Preprint

View full text Add to dashboard Cite

We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficient, it introduces noise in the corresponding estimation of weakly regular parameters, whence we can improve the estimators by local asymptotic Rao-Blackwellization. We call an estimator weakly efficient if it does not admit such improvement. New weakly efficient estimators are presented in linear IV and nonlinear regression models. Simulation of a linear IV model demonstrates how 2SLS and optimal IV estimators are improved.

show abstract

Unbiased Instrumental Variables Estimation Under Known First-Stage Sign

Cited by 10 publications

References 31 publications

Theory of Weak Identification in Semiparametric Models

Theory of Weak Identification in Semiparametric Models

Inferring Treatment Effects After Testing Instrument Strength in Linear Models

Theory of Weak Identification in Semiparametric Models

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